Mathematical analysis for nonlinear partial differential equations with singular solutions

本研究的目的是求解非线性偏微分方程，其解随时间具有奇异性。奇异点的候选点是谐波映射中的缺陷、金兹堡-朗道问题中的涡旋和自由边界。我们解决了以下问题：(1)在具有自由边界的皂膜振动中，我们已经建立了处理波浪型自由边界问题的方法(2)我们开发了一种数值方法，通过离散Mores流对体积约束条件(3)我们构造了一个带体积约束的双曲方程的弱解(4)我们构造了一个带体积约束的抛物方程的弱解，并显示了解的Hoelder连续性。此外，我们开发了求解并行机的最小化算法，通过离散Morese流。这非常适用于体积约束问题。这对于弱连接并联机器来说也是非常好的，因为它使用了变分原理的直接方法。最后，我们要特别感谢所有参与这个项目的人。

The aim of this research was to solve nonlinear Partial Differential Equations whose solution is expected to have singularities depending on time. The candidates of singularities are defects in harmonic mapping, vortex in Ginzburg-Landau problem and free boundaries.We have solved the following problems;(1)On a Soap film vibration with free boundary, we have established the method to treat wave type free boundary problems(2)We developed a numerical method via the discrete Mores flow for volume constraint conditions(3)We constructed a weak solution to a hyperbolic equation with volume constraint(4)We constructed a weak solution to a parabolic equation with volume constraint and showing Hoelder continuity of thesolutionMoreover we have developed solvers for parallel machine with minimizing algorithm via the discrete Morese flows. This works very well especially for volume constraint problems. This is also very nice for a weak connected parallel machines, because it uses direct method of variational principle.Finally, we would like to express pur special thanks to all participants of this project.